LMFDB - Embedded newform 6525.2.a.ca.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,-2,0,10,0,0,-1,-9,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 $ x^9 - 2x^8 - 12x^7 + 21x^6 + 48x^5 - 68x^4 - 73x^3 + 66x^2 + 40x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.23642$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23642 q^{2} -0.471260 q^{4} +3.27673 q^{7} +3.05552 q^{8} +2.30259 q^{11} -5.57932 q^{13} -4.05142 q^{14} -2.83539 q^{16} +1.94031 q^{17} -3.59158 q^{19} -2.84698 q^{22} +1.66263 q^{23} +6.89840 q^{26} -1.54419 q^{28} +1.00000 q^{29} +1.70877 q^{31} -2.60530 q^{32} -2.39904 q^{34} +9.16633 q^{37} +4.44071 q^{38} -8.54038 q^{41} +3.56165 q^{43} -1.08512 q^{44} -2.05571 q^{46} -11.5640 q^{47} +3.73698 q^{49} +2.62931 q^{52} -9.66312 q^{53} +10.0121 q^{56} -1.23642 q^{58} -9.83385 q^{59} +5.42714 q^{61} -2.11276 q^{62} +8.89204 q^{64} -5.20114 q^{67} -0.914391 q^{68} -6.02596 q^{71} -15.5143 q^{73} -11.3334 q^{74} +1.69257 q^{76} +7.54498 q^{77} +12.2465 q^{79} +10.5595 q^{82} -10.7179 q^{83} -4.40370 q^{86} +7.03562 q^{88} -2.53756 q^{89} -18.2820 q^{91} -0.783531 q^{92} +14.2980 q^{94} +5.89998 q^{97} -4.62048 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38}+ \cdots - 51 q^{98}+O(q^{100}) $ 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8 + 2 * q^11 - q^13 - 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 - 3 * q^22 - 16 * q^23 + 6 * q^26 + 4 * q^28 + 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 - 10 * q^41 - 3 * q^43 - 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 + 9 * q^52 - 22 * q^53 + 22 * q^56 - 2 * q^58 + 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 - 5 * q^67 - 39 * q^68 + 10 * q^73 - 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 + 8 * q^82 - 46 * q^83 + 28 * q^86 - 2 * q^88 + 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 - 7 * q^97 - 51 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.23642	−0.874282	−0.437141	−	0.899393i	$-0.644009\pi$
$2$	−1.23642	−0.874282	−0.437141	+	0.899393i	$0.644009\pi$
$3$	0	0
$3$	0	0
$4$	−0.471260	−0.235630
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.27673	1.23849	0.619244	−	0.785198i	$-0.287439\pi$
$7$	3.27673	1.23849	0.619244	+	0.785198i	$0.287439\pi$
$8$	3.05552	1.08029
$9$	0	0
$10$	0	0
$11$	2.30259	0.694258	0.347129	−	0.937817i	$-0.387157\pi$
$11$	2.30259	0.694258	0.347129	+	0.937817i	$0.387157\pi$
$12$	0	0
$13$	−5.57932	−1.54743	−0.773713	−	0.633536i	$-0.781603\pi$
$13$	−5.57932	−1.54743	−0.773713	+	0.633536i	$0.781603\pi$
$14$	−4.05142	−1.08279
$15$	0	0
$16$	−2.83539	−0.708848
$17$	1.94031	0.470594	0.235297	−	0.971923i	$-0.424394\pi$
$17$	1.94031	0.470594	0.235297	+	0.971923i	$0.424394\pi$
$18$	0	0
$19$	−3.59158	−0.823965	−0.411982	−	0.911192i	$-0.635164\pi$
$19$	−3.59158	−0.823965	−0.411982	+	0.911192i	$0.635164\pi$
$20$	0	0
$21$	0	0
$22$	−2.84698	−0.606977
$23$	1.66263	0.346682	0.173341	−	0.984862i	$-0.444544\pi$
$23$	1.66263	0.346682	0.173341	+	0.984862i	$0.444544\pi$
$24$	0	0
$25$	0	0
$26$	6.89840	1.35289
$27$	0	0
$28$	−1.54419	−0.291825
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.70877	0.306903	0.153452	−	0.988156i	$-0.450961\pi$
$31$	1.70877	0.306903	0.153452	+	0.988156i	$0.450961\pi$
$32$	−2.60530	−0.460556
$33$	0	0
$34$	−2.39904	−0.411432
$35$	0	0
$36$	0	0
$37$	9.16633	1.50693	0.753467	−	0.657485i	$-0.228380\pi$
$37$	9.16633	1.50693	0.753467	+	0.657485i	$0.228380\pi$
$38$	4.44071	0.720378
$39$	0	0
$40$	0	0
$41$	−8.54038	−1.33378	−0.666892	−	0.745154i	$-0.732376\pi$
$41$	−8.54038	−1.33378	−0.666892	+	0.745154i	$0.732376\pi$
$42$	0	0
$43$	3.56165	0.543146	0.271573	−	0.962418i	$-0.412456\pi$
$43$	3.56165	0.543146	0.271573	+	0.962418i	$0.412456\pi$
$44$	−1.08512	−0.163588
$45$	0	0
$46$	−2.05571	−0.303098
$47$	−11.5640	−1.68678	−0.843391	−	0.537300i	$-0.819444\pi$
$47$	−11.5640	−1.68678	−0.843391	+	0.537300i	$0.819444\pi$
$48$	0	0
$49$	3.73698	0.533854
$50$	0	0
$51$	0	0
$52$	2.62931	0.364620
$53$	−9.66312	−1.32733	−0.663666	−	0.748029i	$-0.731000\pi$
$53$	−9.66312	−1.32733	−0.663666	+	0.748029i	$0.731000\pi$
$54$	0	0
$55$	0	0
$56$	10.0121	1.33793
$57$	0	0
$58$	−1.23642	−0.162350
$59$	−9.83385	−1.28026	−0.640129	−	0.768268i	$-0.721119\pi$
$59$	−9.83385	−1.28026	−0.640129	+	0.768268i	$0.721119\pi$
$60$	0	0
$61$	5.42714	0.694874	0.347437	−	0.937703i	$-0.387052\pi$
$61$	5.42714	0.694874	0.347437	+	0.937703i	$0.387052\pi$
$62$	−2.11276	−0.268320
$63$	0	0
$64$	8.89204	1.11150
$65$	0	0
$66$	0	0
$67$	−5.20114	−0.635421	−0.317710	−	0.948188i	$-0.602914\pi$
$67$	−5.20114	−0.635421	−0.317710	+	0.948188i	$0.602914\pi$
$68$	−0.914391	−0.110886
$69$	0	0
$70$	0	0
$71$	−6.02596	−0.715150	−0.357575	−	0.933884i	$-0.616396\pi$
$71$	−6.02596	−0.715150	−0.357575	+	0.933884i	$0.616396\pi$
$72$	0	0
$73$	−15.5143	−1.81581	−0.907905	−	0.419177i	$-0.862319\pi$
$73$	−15.5143	−1.81581	−0.907905	+	0.419177i	$0.862319\pi$
$74$	−11.3334	−1.31749
$75$	0	0
$76$	1.69257	0.194151
$77$	7.54498	0.859830
$78$	0	0
$79$	12.2465	1.37784	0.688922	−	0.724835i	$-0.258084\pi$
$79$	12.2465	1.37784	0.688922	+	0.724835i	$0.258084\pi$
$80$	0	0
$81$	0	0
$82$	10.5595	1.16610
$83$	−10.7179	−1.17644	−0.588221	−	0.808700i	$-0.700172\pi$
$83$	−10.7179	−1.17644	−0.588221	+	0.808700i	$0.700172\pi$
$84$	0	0
$85$	0	0
$86$	−4.40370	−0.474863
$87$	0	0
$88$	7.03562	0.749999
$89$	−2.53756	−0.268981	−0.134491	−	0.990915i	$-0.542940\pi$
$89$	−2.53756	−0.268981	−0.134491	+	0.990915i	$0.542940\pi$
$90$	0	0
$91$	−18.2820	−1.91647
$92$	−0.783531	−0.0816887
$93$	0	0
$94$	14.2980	1.47472
$95$	0	0
$96$	0	0
$97$	5.89998	0.599053	0.299526	−	0.954088i	$-0.403171\pi$
$97$	5.89998	0.599053	0.299526	+	0.954088i	$0.403171\pi$
$98$	−4.62048	−0.466739
$99$	0	0
$100$	0	0
$101$	−4.33789	−0.431636	−0.215818	−	0.976434i	$-0.569242\pi$
$101$	−4.33789	−0.431636	−0.215818	+	0.976434i	$0.569242\pi$
$102$	0	0
$103$	−7.62183	−0.751001	−0.375501	−	0.926822i	$-0.622529\pi$
$103$	−7.62183	−0.751001	−0.375501	+	0.926822i	$0.622529\pi$
$104$	−17.0477	−1.67167
$105$	0	0
$106$	11.9477	1.16046
$107$	−4.19703	−0.405742	−0.202871	−	0.979205i	$-0.565027\pi$
$107$	−4.19703	−0.405742	−0.202871	+	0.979205i	$0.565027\pi$
$108$	0	0
$109$	4.02677	0.385695	0.192847	−	0.981229i	$-0.438228\pi$
$109$	4.02677	0.385695	0.192847	+	0.981229i	$0.438228\pi$
$110$	0	0
$111$	0	0
$112$	−9.29082	−0.877900
$113$	−7.40527	−0.696629	−0.348314	−	0.937378i	$-0.613246\pi$
$113$	−7.40527	−0.696629	−0.348314	+	0.937378i	$0.613246\pi$
$114$	0	0
$115$	0	0
$116$	−0.471260	−0.0437554
$117$	0	0
$118$	12.1588	1.11931
$119$	6.35788	0.582826
$120$	0	0
$121$	−5.69807	−0.518006
$122$	−6.71024	−0.607516
$123$	0	0
$124$	−0.805274	−0.0723157
$125$	0	0
$126$	0	0
$127$	−12.9643	−1.15040	−0.575199	−	0.818014i	$-0.695075\pi$
$127$	−12.9643	−1.15040	−0.575199	+	0.818014i	$0.695075\pi$
$128$	−5.78371	−0.511213
$129$	0	0
$130$	0	0
$131$	17.6941	1.54594	0.772971	−	0.634442i	$-0.218770\pi$
$131$	17.6941	1.54594	0.772971	+	0.634442i	$0.218770\pi$
$132$	0	0
$133$	−11.7686	−1.02047
$134$	6.43081	0.555537
$135$	0	0
$136$	5.92866	0.508378
$137$	22.5543	1.92694	0.963471	−	0.267813i	$-0.0863010\pi$
$137$	22.5543	1.92694	0.963471	+	0.267813i	$0.0863010\pi$
$138$	0	0
$139$	−3.20075	−0.271484	−0.135742	−	0.990744i	$-0.543342\pi$
$139$	−3.20075	−0.271484	−0.135742	+	0.990744i	$0.543342\pi$
$140$	0	0
$141$	0	0
$142$	7.45064	0.625243
$143$	−12.8469	−1.07431
$144$	0	0
$145$	0	0
$146$	19.1822	1.58753
$147$	0	0
$148$	−4.31973	−0.355079
$149$	−0.612365	−0.0501669	−0.0250835	−	0.999685i	$-0.507985\pi$
$149$	−0.612365	−0.0501669	−0.0250835	+	0.999685i	$0.507985\pi$
$150$	0	0
$151$	12.4363	1.01205	0.506026	−	0.862518i	$-0.331114\pi$
$151$	12.4363	1.01205	0.506026	+	0.862518i	$0.331114\pi$
$152$	−10.9741	−0.890121
$153$	0	0
$154$	−9.32878	−0.751734
$155$	0	0
$156$	0	0
$157$	0.378638	0.0302186	0.0151093	−	0.999886i	$-0.495190\pi$
$157$	0.378638	0.0302186	0.0151093	+	0.999886i	$0.495190\pi$
$158$	−15.1419	−1.20463
$159$	0	0
$160$	0	0
$161$	5.44799	0.429362
$162$	0	0
$163$	−2.54446	−0.199297	−0.0996487	−	0.995023i	$-0.531772\pi$
$163$	−2.54446	−0.199297	−0.0996487	+	0.995023i	$0.531772\pi$
$164$	4.02474	0.314280
$165$	0	0
$166$	13.2519	1.02854
$167$	10.7799	0.834175	0.417088	−	0.908866i	$-0.363051\pi$
$167$	10.7799	0.834175	0.417088	+	0.908866i	$0.363051\pi$
$168$	0	0
$169$	18.1289	1.39453
$170$	0	0
$171$	0	0
$172$	−1.67846	−0.127982
$173$	−23.1221	−1.75794	−0.878970	−	0.476877i	$-0.841769\pi$
$173$	−23.1221	−1.75794	−0.878970	+	0.476877i	$0.841769\pi$
$174$	0	0
$175$	0	0
$176$	−6.52875	−0.492123
$177$	0	0
$178$	3.13750	0.235165
$179$	23.7081	1.77203	0.886014	−	0.463659i	$-0.153464\pi$
$179$	23.7081	1.77203	0.886014	+	0.463659i	$0.153464\pi$
$180$	0	0
$181$	11.6809	0.868233	0.434116	−	0.900857i	$-0.357061\pi$
$181$	11.6809	0.868233	0.434116	+	0.900857i	$0.357061\pi$
$182$	22.6042	1.67554
$183$	0	0
$184$	5.08020	0.374517
$185$	0	0
$186$	0	0
$187$	4.46774	0.326714
$188$	5.44965	0.397457
$189$	0	0
$190$	0	0
$191$	8.68267	0.628256	0.314128	−	0.949381i	$-0.398288\pi$
$191$	8.68267	0.628256	0.314128	+	0.949381i	$0.398288\pi$
$192$	0	0
$193$	2.73727	0.197033	0.0985164	−	0.995135i	$-0.468590\pi$
$193$	2.73727	0.197033	0.0985164	+	0.995135i	$0.468590\pi$
$194$	−7.29487	−0.523741
$195$	0	0
$196$	−1.76109	−0.125792
$197$	−26.2457	−1.86993	−0.934964	−	0.354741i	$-0.884569\pi$
$197$	−26.2457	−1.86993	−0.934964	+	0.354741i	$0.884569\pi$
$198$	0	0
$199$	9.94800	0.705195	0.352597	−	0.935775i	$-0.385299\pi$
$199$	9.94800	0.705195	0.352597	+	0.935775i	$0.385299\pi$
$200$	0	0
$201$	0	0
$202$	5.36347	0.377372
$203$	3.27673	0.229982
$204$	0	0
$205$	0	0
$206$	9.42380	0.656587
$207$	0	0
$208$	15.8196	1.09689
$209$	−8.26994	−0.572044
$210$	0	0
$211$	7.76180	0.534344	0.267172	−	0.963649i	$-0.413911\pi$
$211$	7.76180	0.534344	0.267172	+	0.963649i	$0.413911\pi$
$212$	4.55384	0.312759
$213$	0	0
$214$	5.18930	0.354733
$215$	0	0
$216$	0	0
$217$	5.59917	0.380096
$218$	−4.97879	−0.337206
$219$	0	0
$220$	0	0
$221$	−10.8256	−0.728210
$222$	0	0
$223$	9.63976	0.645526	0.322763	−	0.946480i	$-0.395388\pi$
$223$	9.63976	0.645526	0.322763	+	0.946480i	$0.395388\pi$
$224$	−8.53687	−0.570394
$225$	0	0
$226$	9.15603	0.609050
$227$	−9.21807	−0.611825	−0.305913	−	0.952060i	$-0.598962\pi$
$227$	−9.21807	−0.611825	−0.305913	+	0.952060i	$0.598962\pi$
$228$	0	0
$229$	−6.89586	−0.455691	−0.227846	−	0.973697i	$-0.573168\pi$
$229$	−6.89586	−0.455691	−0.227846	+	0.973697i	$0.573168\pi$
$230$	0	0
$231$	0	0
$232$	3.05552	0.200605
$233$	13.6229	0.892466	0.446233	−	0.894917i	$-0.352765\pi$
$233$	13.6229	0.892466	0.446233	+	0.894917i	$0.352765\pi$
$234$	0	0
$235$	0	0
$236$	4.63430	0.301667
$237$	0	0
$238$	−7.86102	−0.509554
$239$	−0.749144	−0.0484581	−0.0242290	−	0.999706i	$-0.507713\pi$
$239$	−0.749144	−0.0484581	−0.0242290	+	0.999706i	$0.507713\pi$
$240$	0	0
$241$	−12.5380	−0.807645	−0.403822	−	0.914837i	$-0.632319\pi$
$241$	−12.5380	−0.807645	−0.403822	+	0.914837i	$0.632319\pi$
$242$	7.04522	0.452884
$243$	0	0
$244$	−2.55760	−0.163733
$245$	0	0
$246$	0	0
$247$	20.0386	1.27503
$248$	5.22117	0.331545
$249$	0	0
$250$	0	0
$251$	11.7441	0.741281	0.370640	−	0.928776i	$-0.379138\pi$
$251$	11.7441	0.741281	0.370640	+	0.928776i	$0.379138\pi$
$252$	0	0
$253$	3.82836	0.240687
$254$	16.0294	1.00577
$255$	0	0
$256$	−10.6330	−0.664560
$257$	−13.9304	−0.868955	−0.434477	−	0.900683i	$-0.643067\pi$
$257$	−13.9304	−0.868955	−0.434477	+	0.900683i	$0.643067\pi$
$258$	0	0
$259$	30.0356	1.86632
$260$	0	0
$261$	0	0
$262$	−21.8774	−1.35159
$263$	−25.8059	−1.59126	−0.795631	−	0.605781i	$-0.792861\pi$
$263$	−25.8059	−1.59126	−0.795631	+	0.605781i	$0.792861\pi$
$264$	0	0
$265$	0	0
$266$	14.5510	0.892180
$267$	0	0
$268$	2.45109	0.149724
$269$	−11.4590	−0.698667	−0.349333	−	0.936998i	$-0.613592\pi$
$269$	−11.4590	−0.698667	−0.349333	+	0.936998i	$0.613592\pi$
$270$	0	0
$271$	−21.4783	−1.30472	−0.652358	−	0.757911i	$-0.726220\pi$
$271$	−21.4783	−1.30472	−0.652358	+	0.757911i	$0.726220\pi$
$272$	−5.50154	−0.333580
$273$	0	0
$274$	−27.8866	−1.68469
$275$	0	0
$276$	0	0
$277$	7.16907	0.430747	0.215374	−	0.976532i	$-0.430903\pi$
$277$	7.16907	0.430747	0.215374	+	0.976532i	$0.430903\pi$
$278$	3.95748	0.237354
$279$	0	0
$280$	0	0
$281$	14.0042	0.835420	0.417710	−	0.908580i	$-0.362833\pi$
$281$	14.0042	0.835420	0.417710	+	0.908580i	$0.362833\pi$
$282$	0	0
$283$	10.3513	0.615324	0.307662	−	0.951496i	$-0.400453\pi$
$283$	10.3513	0.615324	0.307662	+	0.951496i	$0.400453\pi$
$284$	2.83980	0.168511
$285$	0	0
$286$	15.8842	0.939253
$287$	−27.9846	−1.65188
$288$	0	0
$289$	−13.2352	−0.778541
$290$	0	0
$291$	0	0
$292$	7.31127	0.427859
$293$	−26.6063	−1.55435	−0.777177	−	0.629282i	$-0.783349\pi$
$293$	−26.6063	−1.55435	−0.777177	+	0.629282i	$0.783349\pi$
$294$	0	0
$295$	0	0
$296$	28.0079	1.62793
$297$	0	0
$298$	0.757142	0.0438601
$299$	−9.27634	−0.536465
$300$	0	0
$301$	11.6706	0.672680
$302$	−15.3765	−0.884819
$303$	0	0
$304$	10.1835	0.584066
$305$	0	0
$306$	0	0
$307$	29.2808	1.67114	0.835571	−	0.549383i	$-0.185137\pi$
$307$	29.2808	1.67114	0.835571	+	0.549383i	$0.185137\pi$
$308$	−3.55565	−0.202602
$309$	0	0
$310$	0	0
$311$	−5.03104	−0.285284	−0.142642	−	0.989774i	$-0.545560\pi$
$311$	−5.03104	−0.285284	−0.142642	+	0.989774i	$0.545560\pi$
$312$	0	0
$313$	−27.5875	−1.55934	−0.779670	−	0.626191i	$-0.784613\pi$
$313$	−27.5875	−1.55934	−0.779670	+	0.626191i	$0.784613\pi$
$314$	−0.468157	−0.0264196
$315$	0	0
$316$	−5.77131	−0.324662
$317$	7.70663	0.432848	0.216424	−	0.976300i	$-0.430561\pi$
$317$	7.70663	0.432848	0.216424	+	0.976300i	$0.430561\pi$
$318$	0	0
$319$	2.30259	0.128920
$320$	0	0
$321$	0	0
$322$	−6.73601	−0.375383
$323$	−6.96878	−0.387753
$324$	0	0
$325$	0	0
$326$	3.14602	0.174242
$327$	0	0
$328$	−26.0953	−1.44087
$329$	−37.8921	−2.08906
$330$	0	0
$331$	18.9943	1.04402	0.522010	−	0.852940i	$-0.325182\pi$
$331$	18.9943	1.04402	0.522010	+	0.852940i	$0.325182\pi$
$332$	5.05092	0.277205
$333$	0	0
$334$	−13.3285	−0.729305
$335$	0	0
$336$	0	0
$337$	−20.1983	−1.10027	−0.550136	−	0.835075i	$-0.685424\pi$
$337$	−20.1983	−1.10027	−0.550136	+	0.835075i	$0.685424\pi$
$338$	−22.4149	−1.21921
$339$	0	0
$340$	0	0
$341$	3.93459	0.213070
$342$	0	0
$343$	−10.6921	−0.577317
$344$	10.8827	0.586755
$345$	0	0
$346$	28.5887	1.53694
$347$	−23.7995	−1.27763	−0.638813	−	0.769362i	$-0.720574\pi$
$347$	−23.7995	−1.27763	−0.638813	+	0.769362i	$0.720574\pi$
$348$	0	0
$349$	−22.8554	−1.22342	−0.611710	−	0.791082i	$-0.709518\pi$
$349$	−22.8554	−1.22342	−0.611710	+	0.791082i	$0.709518\pi$
$350$	0	0
$351$	0	0
$352$	−5.99894	−0.319745
$353$	17.5338	0.933231	0.466615	−	0.884460i	$-0.345473\pi$
$353$	17.5338	0.933231	0.466615	+	0.884460i	$0.345473\pi$
$354$	0	0
$355$	0	0
$356$	1.19585	0.0633801
$357$	0	0
$358$	−29.3132	−1.54925
$359$	26.6594	1.40703	0.703513	−	0.710682i	$-0.251613\pi$
$359$	26.6594	1.40703	0.703513	+	0.710682i	$0.251613\pi$
$360$	0	0
$361$	−6.10055	−0.321082
$362$	−14.4425	−0.759081
$363$	0	0
$364$	8.61556	0.451578
$365$	0	0
$366$	0	0
$367$	−22.5869	−1.17902	−0.589512	−	0.807760i	$-0.700680\pi$
$367$	−22.5869	−1.17902	−0.589512	+	0.807760i	$0.700680\pi$
$368$	−4.71420	−0.245745
$369$	0	0
$370$	0	0
$371$	−31.6634	−1.64388
$372$	0	0
$373$	11.0978	0.574623	0.287311	−	0.957837i	$-0.407239\pi$
$373$	11.0978	0.574623	0.287311	+	0.957837i	$0.407239\pi$
$374$	−5.52402	−0.285640
$375$	0	0
$376$	−35.3340	−1.82221
$377$	−5.57932	−0.287350
$378$	0	0
$379$	−28.2474	−1.45097	−0.725485	−	0.688238i	$-0.758384\pi$
$379$	−28.2474	−1.45097	−0.725485	+	0.688238i	$0.758384\pi$
$380$	0	0
$381$	0	0
$382$	−10.7354	−0.549273
$383$	−7.24537	−0.370221	−0.185110	−	0.982718i	$-0.559264\pi$
$383$	−7.24537	−0.370221	−0.185110	+	0.982718i	$0.559264\pi$
$384$	0	0
$385$	0	0
$386$	−3.38442	−0.172262
$387$	0	0
$388$	−2.78043	−0.141155
$389$	−34.4482	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$389$	−34.4482	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$390$	0	0
$391$	3.22601	0.163147
$392$	11.4184	0.576717
$393$	0	0
$394$	32.4508	1.63485
$395$	0	0
$396$	0	0
$397$	−13.6412	−0.684632	−0.342316	−	0.939585i	$-0.611211\pi$
$397$	−13.6412	−0.684632	−0.342316	+	0.939585i	$0.611211\pi$
$398$	−12.2999	−0.616540
$399$	0	0
$400$	0	0
$401$	−10.3741	−0.518059	−0.259029	−	0.965869i	$-0.583403\pi$
$401$	−10.3741	−0.518059	−0.259029	+	0.965869i	$0.583403\pi$
$402$	0	0
$403$	−9.53376	−0.474910
$404$	2.04428	0.101707
$405$	0	0
$406$	−4.05142	−0.201069
$407$	21.1063	1.04620
$408$	0	0
$409$	38.5394	1.90565	0.952825	−	0.303520i	$-0.0981620\pi$
$409$	38.5394	1.90565	0.952825	+	0.303520i	$0.0981620\pi$
$410$	0	0
$411$	0	0
$412$	3.59187	0.176959
$413$	−32.2229	−1.58558
$414$	0	0
$415$	0	0
$416$	14.5358	0.712677
$417$	0	0
$418$	10.2251	0.500128
$419$	−9.87578	−0.482464	−0.241232	−	0.970468i	$-0.577551\pi$
$419$	−9.87578	−0.482464	−0.241232	+	0.970468i	$0.577551\pi$
$420$	0	0
$421$	−10.2857	−0.501292	−0.250646	−	0.968079i	$-0.580643\pi$
$421$	−10.2857	−0.501292	−0.250646	+	0.968079i	$0.580643\pi$
$422$	−9.59686	−0.467168
$423$	0	0
$424$	−29.5259	−1.43390
$425$	0	0
$426$	0	0
$427$	17.7833	0.860594
$428$	1.97789	0.0956051
$429$	0	0
$430$	0	0
$431$	15.2705	0.735553	0.367776	−	0.929914i	$-0.380119\pi$
$431$	15.2705	0.735553	0.367776	+	0.929914i	$0.380119\pi$
$432$	0	0
$433$	−13.8370	−0.664966	−0.332483	−	0.943109i	$-0.607886\pi$
$433$	−13.8370	−0.664966	−0.332483	+	0.943109i	$0.607886\pi$
$434$	−6.92294	−0.332312
$435$	0	0
$436$	−1.89766	−0.0908813
$437$	−5.97146	−0.285654
$438$	0	0
$439$	−6.48836	−0.309673	−0.154836	−	0.987940i	$-0.549485\pi$
$439$	−6.48836	−0.309673	−0.154836	+	0.987940i	$0.549485\pi$
$440$	0	0
$441$	0	0
$442$	13.3850	0.636661
$443$	−22.1026	−1.05012	−0.525062	−	0.851064i	$-0.675958\pi$
$443$	−22.1026	−1.05012	−0.525062	+	0.851064i	$0.675958\pi$
$444$	0	0
$445$	0	0
$446$	−11.9188	−0.564372
$447$	0	0
$448$	29.1368	1.37659
$449$	−0.247203	−0.0116662	−0.00583311	−	0.999983i	$-0.501857\pi$
$449$	−0.247203	−0.0116662	−0.00583311	+	0.999983i	$0.501857\pi$
$450$	0	0
$451$	−19.6650	−0.925990
$452$	3.48981	0.164147
$453$	0	0
$454$	11.3974	0.534908
$455$	0	0
$456$	0	0
$457$	35.4153	1.65666	0.828328	−	0.560243i	$-0.189292\pi$
$457$	35.4153	1.65666	0.828328	+	0.560243i	$0.189292\pi$
$458$	8.52619	0.398403
$459$	0	0
$460$	0	0
$461$	−13.5690	−0.631969	−0.315985	−	0.948764i	$-0.602335\pi$
$461$	−13.5690	−0.631969	−0.315985	+	0.948764i	$0.602335\pi$
$462$	0	0
$463$	7.88217	0.366316	0.183158	−	0.983084i	$-0.441368\pi$
$463$	7.88217	0.366316	0.183158	+	0.983084i	$0.441368\pi$
$464$	−2.83539	−0.131630
$465$	0	0
$466$	−16.8437	−0.780268
$467$	−20.8411	−0.964411	−0.482206	−	0.876058i	$-0.660164\pi$
$467$	−20.8411	−0.964411	−0.482206	+	0.876058i	$0.660164\pi$
$468$	0	0
$469$	−17.0428	−0.786961
$470$	0	0
$471$	0	0
$472$	−30.0475	−1.38305
$473$	8.20102	0.377083
$474$	0	0
$475$	0	0
$476$	−2.99622	−0.137331
$477$	0	0
$478$	0.926258	0.0423661
$479$	−28.9598	−1.32321	−0.661604	−	0.749853i	$-0.730124\pi$
$479$	−28.9598	−1.32321	−0.661604	+	0.749853i	$0.730124\pi$
$480$	0	0
$481$	−51.1419	−2.33187
$482$	15.5023	0.706110
$483$	0	0
$484$	2.68527	0.122058
$485$	0	0
$486$	0	0
$487$	23.3354	1.05743	0.528714	−	0.848800i	$-0.322674\pi$
$487$	23.3354	1.05743	0.528714	+	0.848800i	$0.322674\pi$
$488$	16.5827	0.750665
$489$	0	0
$490$	0	0
$491$	25.9951	1.17314	0.586570	−	0.809898i	$-0.300478\pi$
$491$	25.9951	1.17314	0.586570	+	0.809898i	$0.300478\pi$
$492$	0	0
$493$	1.94031	0.0873872
$494$	−24.7762	−1.11473
$495$	0	0
$496$	−4.84502	−0.217548
$497$	−19.7455	−0.885706
$498$	0	0
$499$	−31.9575	−1.43061	−0.715307	−	0.698810i	$-0.753713\pi$
$499$	−31.9575	−1.43061	−0.715307	+	0.698810i	$0.753713\pi$
$500$	0	0
$501$	0	0
$502$	−14.5207	−0.648089
$503$	−4.75172	−0.211869	−0.105934	−	0.994373i	$-0.533783\pi$
$503$	−4.75172	−0.211869	−0.105934	+	0.994373i	$0.533783\pi$
$504$	0	0
$505$	0	0
$506$	−4.73346	−0.210428
$507$	0	0
$508$	6.10957	0.271068
$509$	−10.8427	−0.480595	−0.240297	−	0.970699i	$-0.577245\pi$
$509$	−10.8427	−0.480595	−0.240297	+	0.970699i	$0.577245\pi$
$510$	0	0
$511$	−50.8361	−2.24886
$512$	24.7143	1.09223
$513$	0	0
$514$	17.2239	0.759712
$515$	0	0
$516$	0	0
$517$	−26.6272	−1.17106
$518$	−37.1367	−1.63169
$519$	0	0
$520$	0	0
$521$	−33.8929	−1.48487	−0.742437	−	0.669916i	$-0.766330\pi$
$521$	−33.8929	−1.48487	−0.742437	+	0.669916i	$0.766330\pi$
$522$	0	0
$523$	25.7652	1.12664	0.563318	−	0.826240i	$-0.309525\pi$
$523$	25.7652	1.12664	0.563318	+	0.826240i	$0.309525\pi$
$524$	−8.33853	−0.364270
$525$	0	0
$526$	31.9070	1.39121
$527$	3.31554	0.144427
$528$	0	0
$529$	−20.2357	−0.879812
$530$	0	0
$531$	0	0
$532$	5.54610	0.240454
$533$	47.6496	2.06393
$534$	0	0
$535$	0	0
$536$	−15.8922	−0.686438
$537$	0	0
$538$	14.1681	0.610832
$539$	8.60473	0.370632
$540$	0	0
$541$	4.96888	0.213629	0.106814	−	0.994279i	$-0.465935\pi$
$541$	4.96888	0.213629	0.106814	+	0.994279i	$0.465935\pi$
$542$	26.5563	1.14069
$543$	0	0
$544$	−5.05509	−0.216735
$545$	0	0
$546$	0	0
$547$	−9.03353	−0.386246	−0.193123	−	0.981175i	$-0.561862\pi$
$547$	−9.03353	−0.386246	−0.193123	+	0.981175i	$0.561862\pi$
$548$	−10.6289	−0.454046
$549$	0	0
$550$	0	0
$551$	−3.59158	−0.153006
$552$	0	0
$553$	40.1287	1.70644
$554$	−8.86399	−0.376595
$555$	0	0
$556$	1.50839	0.0639699
$557$	−22.5265	−0.954477	−0.477238	−	0.878774i	$-0.658362\pi$
$557$	−22.5265	−0.954477	−0.477238	+	0.878774i	$0.658362\pi$
$558$	0	0
$559$	−19.8716	−0.840479
$560$	0	0
$561$	0	0
$562$	−17.3151	−0.730393
$563$	−14.0601	−0.592561	−0.296280	−	0.955101i	$-0.595746\pi$
$563$	−14.0601	−0.592561	−0.296280	+	0.955101i	$0.595746\pi$
$564$	0	0
$565$	0	0
$566$	−12.7986	−0.537967
$567$	0	0
$568$	−18.4125	−0.772570
$569$	5.59236	0.234444	0.117222	−	0.993106i	$-0.462601\pi$
$569$	5.59236	0.234444	0.117222	+	0.993106i	$0.462601\pi$
$570$	0	0
$571$	−20.9909	−0.878443	−0.439221	−	0.898379i	$-0.644746\pi$
$571$	−20.9909	−0.878443	−0.439221	+	0.898379i	$0.644746\pi$
$572$	6.05424	0.253140
$573$	0	0
$574$	34.6007	1.44421
$575$	0	0
$576$	0	0
$577$	25.7139	1.07048	0.535242	−	0.844699i	$-0.320221\pi$
$577$	25.7139	1.07048	0.535242	+	0.844699i	$0.320221\pi$
$578$	16.3643	0.680665
$579$	0	0
$580$	0	0
$581$	−35.1197	−1.45701
$582$	0	0
$583$	−22.2502	−0.921510
$584$	−47.4042	−1.96160
$585$	0	0
$586$	32.8966	1.35894
$587$	9.26970	0.382602	0.191301	−	0.981531i	$-0.438729\pi$
$587$	9.26970	0.382602	0.191301	+	0.981531i	$0.438729\pi$
$588$	0	0
$589$	−6.13717	−0.252878
$590$	0	0
$591$	0	0
$592$	−25.9901	−1.06819
$593$	31.5912	1.29729	0.648647	−	0.761089i	$-0.275335\pi$
$593$	31.5912	1.29729	0.648647	+	0.761089i	$0.275335\pi$
$594$	0	0
$595$	0	0
$596$	0.288584	0.0118208
$597$	0	0
$598$	11.4695	0.469022
$599$	13.1545	0.537480	0.268740	−	0.963213i	$-0.413393\pi$
$599$	13.1545	0.537480	0.268740	+	0.963213i	$0.413393\pi$
$600$	0	0
$601$	34.2150	1.39566	0.697829	−	0.716265i	$-0.254150\pi$
$601$	34.2150	1.39566	0.697829	+	0.716265i	$0.254150\pi$
$602$	−14.4297	−0.588113
$603$	0	0
$604$	−5.86073	−0.238470
$605$	0	0
$606$	0	0
$607$	−3.51939	−0.142848	−0.0714238	−	0.997446i	$-0.522754\pi$
$607$	−3.51939	−0.142848	−0.0714238	+	0.997446i	$0.522754\pi$
$608$	9.35714	0.379482
$609$	0	0
$610$	0	0
$611$	64.5193	2.61017
$612$	0	0
$613$	5.90592	0.238538	0.119269	−	0.992862i	$-0.461945\pi$
$613$	5.90592	0.238538	0.119269	+	0.992862i	$0.461945\pi$
$614$	−36.2034	−1.46105
$615$	0	0
$616$	23.0538	0.928866
$617$	−30.3173	−1.22053	−0.610264	−	0.792198i	$-0.708937\pi$
$617$	−30.3173	−1.22053	−0.610264	+	0.792198i	$0.708937\pi$
$618$	0	0
$619$	−23.1044	−0.928646	−0.464323	−	0.885666i	$-0.653702\pi$
$619$	−23.1044	−0.928646	−0.464323	+	0.885666i	$0.653702\pi$
$620$	0	0
$621$	0	0
$622$	6.22049	0.249419
$623$	−8.31491	−0.333130
$624$	0	0
$625$	0	0
$626$	34.1098	1.36330
$627$	0	0
$628$	−0.178437	−0.00712042
$629$	17.7855	0.709155
$630$	0	0
$631$	−27.6228	−1.09965	−0.549824	−	0.835280i	$-0.685305\pi$
$631$	−27.6228	−1.09965	−0.549824	+	0.835280i	$0.685305\pi$
$632$	37.4196	1.48847
$633$	0	0
$634$	−9.52865	−0.378431
$635$	0	0
$636$	0	0
$637$	−20.8498	−0.826099
$638$	−2.84698	−0.112713
$639$	0	0
$640$	0	0
$641$	−10.7286	−0.423756	−0.211878	−	0.977296i	$-0.567958\pi$
$641$	−10.7286	−0.423756	−0.211878	+	0.977296i	$0.567958\pi$
$642$	0	0
$643$	18.0482	0.711751	0.355876	−	0.934533i	$-0.384183\pi$
$643$	18.0482	0.711751	0.355876	+	0.934533i	$0.384183\pi$
$644$	−2.56742	−0.101171
$645$	0	0
$646$	8.61635	0.339006
$647$	−26.9395	−1.05910	−0.529551	−	0.848278i	$-0.677640\pi$
$647$	−26.9395	−1.05910	−0.529551	+	0.848278i	$0.677640\pi$
$648$	0	0
$649$	−22.6433	−0.888829
$650$	0	0
$651$	0	0
$652$	1.19910	0.0469605
$653$	−13.4857	−0.527737	−0.263868	−	0.964559i	$-0.584998\pi$
$653$	−13.4857	−0.527737	−0.263868	+	0.964559i	$0.584998\pi$
$654$	0	0
$655$	0	0
$656$	24.2153	0.945450
$657$	0	0
$658$	46.8507	1.82643
$659$	27.5560	1.07343	0.536716	−	0.843763i	$-0.319665\pi$
$659$	27.5560	1.07343	0.536716	+	0.843763i	$0.319665\pi$
$660$	0	0
$661$	18.1148	0.704583	0.352292	−	0.935890i	$-0.385403\pi$
$661$	18.1148	0.704583	0.352292	+	0.935890i	$0.385403\pi$
$662$	−23.4849	−0.912768
$663$	0	0
$664$	−32.7488	−1.27090
$665$	0	0
$666$	0	0
$667$	1.66263	0.0643772
$668$	−5.08015	−0.196557
$669$	0	0
$670$	0	0
$671$	12.4965	0.482422
$672$	0	0
$673$	45.5751	1.75679	0.878396	−	0.477933i	$-0.158614\pi$
$673$	45.5751	1.75679	0.878396	+	0.477933i	$0.158614\pi$
$674$	24.9736	0.961948
$675$	0	0
$676$	−8.54342	−0.328593
$677$	−10.2385	−0.393497	−0.196748	−	0.980454i	$-0.563038\pi$
$677$	−10.2385	−0.393497	−0.196748	+	0.980454i	$0.563038\pi$
$678$	0	0
$679$	19.3327	0.741920
$680$	0	0
$681$	0	0
$682$	−4.86482	−0.186283
$683$	41.9562	1.60541	0.802705	−	0.596376i	$-0.203393\pi$
$683$	41.9562	1.60541	0.802705	+	0.596376i	$0.203393\pi$
$684$	0	0
$685$	0	0
$686$	13.2199	0.504738
$687$	0	0
$688$	−10.0987	−0.385008
$689$	53.9137	2.05395
$690$	0	0
$691$	50.6633	1.92732	0.963662	−	0.267126i	$-0.0860739\pi$
$691$	50.6633	1.92732	0.963662	+	0.267126i	$0.0860739\pi$
$692$	10.8965	0.414224
$693$	0	0
$694$	29.4263	1.11701
$695$	0	0
$696$	0	0
$697$	−16.5710	−0.627671
$698$	28.2589	1.06961
$699$	0	0
$700$	0	0
$701$	−17.2666	−0.652151	−0.326076	−	0.945344i	$-0.605726\pi$
$701$	−17.2666	−0.652151	−0.326076	+	0.945344i	$0.605726\pi$
$702$	0	0
$703$	−32.9216	−1.24166
$704$	20.4747	0.771671
$705$	0	0
$706$	−21.6792	−0.815907
$707$	−14.2141	−0.534577
$708$	0	0
$709$	−18.3307	−0.688425	−0.344213	−	0.938892i	$-0.611854\pi$
$709$	−18.3307	−0.688425	−0.344213	+	0.938892i	$0.611854\pi$
$710$	0	0
$711$	0	0
$712$	−7.75357	−0.290577
$713$	2.84104	0.106398
$714$	0	0
$715$	0	0
$716$	−11.1727	−0.417543
$717$	0	0
$718$	−32.9622	−1.23014
$719$	10.6931	0.398785	0.199392	−	0.979920i	$-0.436103\pi$
$719$	10.6931	0.398785	0.199392	+	0.979920i	$0.436103\pi$
$720$	0	0
$721$	−24.9747	−0.930107
$722$	7.54286	0.280716
$723$	0	0
$724$	−5.50474	−0.204582
$725$	0	0
$726$	0	0
$727$	5.74209	0.212962	0.106481	−	0.994315i	$-0.466042\pi$
$727$	5.74209	0.212962	0.106481	+	0.994315i	$0.466042\pi$
$728$	−55.8609	−2.07034
$729$	0	0
$730$	0	0
$731$	6.91070	0.255601
$732$	0	0
$733$	−6.03079	−0.222752	−0.111376	−	0.993778i	$-0.535526\pi$
$733$	−6.03079	−0.222752	−0.111376	+	0.993778i	$0.535526\pi$
$734$	27.9269	1.03080
$735$	0	0
$736$	−4.33164	−0.159667
$737$	−11.9761	−0.441146
$738$	0	0
$739$	23.7939	0.875273	0.437637	−	0.899152i	$-0.355816\pi$
$739$	23.7939	0.875273	0.437637	+	0.899152i	$0.355816\pi$
$740$	0	0
$741$	0	0
$742$	39.1494	1.43722
$743$	−24.4388	−0.896574	−0.448287	−	0.893890i	$-0.647966\pi$
$743$	−24.4388	−0.896574	−0.448287	+	0.893890i	$0.647966\pi$
$744$	0	0
$745$	0	0
$746$	−13.7216	−0.502383
$747$	0	0
$748$	−2.10547	−0.0769836
$749$	−13.7525	−0.502507
$750$	0	0
$751$	38.9811	1.42244	0.711220	−	0.702970i	$-0.248143\pi$
$751$	38.9811	1.42244	0.711220	+	0.702970i	$0.248143\pi$
$752$	32.7885	1.19567
$753$	0	0
$754$	6.89840	0.251225
$755$	0	0
$756$	0	0
$757$	6.78690	0.246674	0.123337	−	0.992365i	$-0.460640\pi$
$757$	6.78690	0.246674	0.123337	+	0.992365i	$0.460640\pi$
$758$	34.9257	1.26856
$759$	0	0
$760$	0	0
$761$	−8.25388	−0.299203	−0.149601	−	0.988746i	$-0.547799\pi$
$761$	−8.25388	−0.299203	−0.149601	+	0.988746i	$0.547799\pi$
$762$	0	0
$763$	13.1946	0.477678
$764$	−4.09180	−0.148036
$765$	0	0
$766$	8.95833	0.323678
$767$	54.8662	1.98110
$768$	0	0
$769$	−12.3938	−0.446930	−0.223465	−	0.974712i	$-0.571737\pi$
$769$	−12.3938	−0.446930	−0.223465	+	0.974712i	$0.571737\pi$
$770$	0	0
$771$	0	0
$772$	−1.28997	−0.0464269
$773$	38.1311	1.37148	0.685740	−	0.727846i	$-0.259479\pi$
$773$	38.1311	1.37148	0.685740	+	0.727846i	$0.259479\pi$
$774$	0	0
$775$	0	0
$776$	18.0275	0.647150
$777$	0	0
$778$	42.5925	1.52701
$779$	30.6735	1.09899
$780$	0	0
$781$	−13.8753	−0.496499
$782$	−3.98872	−0.142636
$783$	0	0
$784$	−10.5958	−0.378421
$785$	0	0
$786$	0	0
$787$	−37.5823	−1.33967	−0.669833	−	0.742512i	$-0.733634\pi$
$787$	−37.5823	−1.33967	−0.669833	+	0.742512i	$0.733634\pi$
$788$	12.3686	0.440612
$789$	0	0
$790$	0	0
$791$	−24.2651	−0.862767
$792$	0	0
$793$	−30.2798	−1.07527
$794$	16.8663	0.598561
$795$	0	0
$796$	−4.68810	−0.166165
$797$	14.9045	0.527945	0.263972	−	0.964530i	$-0.414967\pi$
$797$	14.9045	0.527945	0.263972	+	0.964530i	$0.414967\pi$
$798$	0	0
$799$	−22.4377	−0.793790
$800$	0	0
$801$	0	0
$802$	12.8268	0.452930
$803$	−35.7231	−1.26064
$804$	0	0
$805$	0	0
$806$	11.7878	0.415206
$807$	0	0
$808$	−13.2545	−0.466292
$809$	−12.5838	−0.442424	−0.221212	−	0.975226i	$-0.571001\pi$
$809$	−12.5838	−0.442424	−0.221212	+	0.975226i	$0.571001\pi$
$810$	0	0
$811$	−27.9894	−0.982841	−0.491420	−	0.870922i	$-0.663522\pi$
$811$	−27.9894	−0.982841	−0.491420	+	0.870922i	$0.663522\pi$
$812$	−1.54419	−0.0541906
$813$	0	0
$814$	−26.0963	−0.914675
$815$	0	0
$816$	0	0
$817$	−12.7919	−0.447533
$818$	−47.6509	−1.66608
$819$	0	0
$820$	0	0
$821$	−1.20609	−0.0420929	−0.0210464	−	0.999778i	$-0.506700\pi$
$821$	−1.20609	−0.0420929	−0.0210464	+	0.999778i	$0.506700\pi$
$822$	0	0
$823$	25.7664	0.898162	0.449081	−	0.893491i	$-0.351751\pi$
$823$	25.7664	0.898162	0.449081	+	0.893491i	$0.351751\pi$
$824$	−23.2887	−0.811299
$825$	0	0
$826$	39.8411	1.38625
$827$	−56.1443	−1.95233	−0.976165	−	0.217029i	$-0.930363\pi$
$827$	−56.1443	−1.95233	−0.976165	+	0.217029i	$0.930363\pi$
$828$	0	0
$829$	−14.6534	−0.508935	−0.254467	−	0.967081i	$-0.581900\pi$
$829$	−14.6534	−0.508935	−0.254467	+	0.967081i	$0.581900\pi$
$830$	0	0
$831$	0	0
$832$	−49.6116	−1.71997
$833$	7.25089	0.251229
$834$	0	0
$835$	0	0
$836$	3.89730	0.134791
$837$	0	0
$838$	12.2106	0.421809
$839$	8.72873	0.301349	0.150675	−	0.988583i	$-0.451855\pi$
$839$	8.72873	0.301349	0.150675	+	0.988583i	$0.451855\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	12.7174	0.438271
$843$	0	0
$844$	−3.65783	−0.125908
$845$	0	0
$846$	0	0
$847$	−18.6710	−0.641545
$848$	27.3987	0.940876
$849$	0	0
$850$	0	0
$851$	15.2402	0.522427
$852$	0	0
$853$	42.3533	1.45015	0.725075	−	0.688670i	$-0.241805\pi$
$853$	42.3533	1.45015	0.725075	+	0.688670i	$0.241805\pi$
$854$	−21.9876	−0.752402
$855$	0	0
$856$	−12.8241	−0.438319
$857$	−49.8924	−1.70429	−0.852145	−	0.523305i	$-0.824699\pi$
$857$	−49.8924	−1.70429	−0.852145	+	0.523305i	$0.824699\pi$
$858$	0	0
$859$	−33.4585	−1.14159	−0.570795	−	0.821092i	$-0.693365\pi$
$859$	−33.4585	−1.14159	−0.570795	+	0.821092i	$0.693365\pi$
$860$	0	0
$861$	0	0
$862$	−18.8807	−0.643081
$863$	−36.8016	−1.25274	−0.626371	−	0.779525i	$-0.715460\pi$
$863$	−36.8016	−1.25274	−0.626371	+	0.779525i	$0.715460\pi$
$864$	0	0
$865$	0	0
$866$	17.1084	0.581368
$867$	0	0
$868$	−2.63867	−0.0895622
$869$	28.1988	0.956579
$870$	0	0
$871$	29.0189	0.983267
$872$	12.3039	0.416662
$873$	0	0
$874$	7.38325	0.249742
$875$	0	0
$876$	0	0
$877$	19.7336	0.666358	0.333179	−	0.942864i	$-0.391879\pi$
$877$	19.7336	0.666358	0.333179	+	0.942864i	$0.391879\pi$
$878$	8.02235	0.270741
$879$	0	0
$880$	0	0
$881$	38.2999	1.29036	0.645179	−	0.764032i	$-0.276783\pi$
$881$	38.2999	1.29036	0.645179	+	0.764032i	$0.276783\pi$
$882$	0	0
$883$	−32.9689	−1.10949	−0.554745	−	0.832020i	$-0.687184\pi$
$883$	−32.9689	−1.10949	−0.554745	+	0.832020i	$0.687184\pi$
$884$	5.10169	0.171588
$885$	0	0
$886$	27.3281	0.918105
$887$	15.0022	0.503724	0.251862	−	0.967763i	$-0.418957\pi$
$887$	15.0022	0.503724	0.251862	+	0.967763i	$0.418957\pi$
$888$	0	0
$889$	−42.4806	−1.42475
$890$	0	0
$891$	0	0
$892$	−4.54284	−0.152105
$893$	41.5330	1.38985
$894$	0	0
$895$	0	0
$896$	−18.9517	−0.633131
$897$	0	0
$898$	0.305647	0.0101996
$899$	1.70877	0.0569905
$900$	0	0
$901$	−18.7494	−0.624634
$902$	24.3143	0.809577
$903$	0	0
$904$	−22.6269	−0.752561
$905$	0	0
$906$	0	0
$907$	45.4276	1.50840	0.754199	−	0.656646i	$-0.228026\pi$
$907$	45.4276	1.50840	0.754199	+	0.656646i	$0.228026\pi$
$908$	4.34411	0.144164
$909$	0	0
$910$	0	0
$911$	59.7103	1.97829	0.989146	−	0.146937i	$-0.0469415\pi$
$911$	59.7103	1.97829	0.989146	+	0.146937i	$0.0469415\pi$
$912$	0	0
$913$	−24.6790	−0.816755
$914$	−43.7882	−1.44839
$915$	0	0
$916$	3.24975	0.107375
$917$	57.9789	1.91463
$918$	0	0
$919$	−1.59828	−0.0527224	−0.0263612	−	0.999652i	$-0.508392\pi$
$919$	−1.59828	−0.0527224	−0.0263612	+	0.999652i	$0.508392\pi$
$920$	0	0
$921$	0	0
$922$	16.7770	0.552520
$923$	33.6208	1.10664
$924$	0	0
$925$	0	0
$926$	−9.74569	−0.320263
$927$	0	0
$928$	−2.60530	−0.0855231
$929$	35.5116	1.16510	0.582549	−	0.812795i	$-0.302055\pi$
$929$	35.5116	1.16510	0.582549	+	0.812795i	$0.302055\pi$
$930$	0	0
$931$	−13.4216	−0.439877
$932$	−6.41994	−0.210292
$933$	0	0
$934$	25.7684	0.843168
$935$	0	0
$936$	0	0
$937$	−52.7492	−1.72324	−0.861621	−	0.507552i	$-0.830551\pi$
$937$	−52.7492	−1.72324	−0.861621	+	0.507552i	$0.830551\pi$
$938$	21.0720	0.688026
$939$	0	0
$940$	0	0
$941$	−27.2935	−0.889743	−0.444872	−	0.895594i	$-0.646751\pi$
$941$	−27.2935	−0.889743	−0.444872	+	0.895594i	$0.646751\pi$
$942$	0	0
$943$	−14.1995	−0.462399
$944$	27.8828	0.907508
$945$	0	0
$946$	−10.1399	−0.329677
$947$	−21.1187	−0.686266	−0.343133	−	0.939287i	$-0.611488\pi$
$947$	−21.1187	−0.686266	−0.343133	+	0.939287i	$0.611488\pi$
$948$	0	0
$949$	86.5592	2.80983
$950$	0	0
$951$	0	0
$952$	19.4266	0.629621
$953$	−3.73049	−0.120842	−0.0604211	−	0.998173i	$-0.519244\pi$
$953$	−3.73049	−0.120842	−0.0604211	+	0.998173i	$0.519244\pi$
$954$	0	0
$955$	0	0
$956$	0.353042	0.0114182
$957$	0	0
$958$	35.8066	1.15686
$959$	73.9043	2.38650
$960$	0	0
$961$	−28.0801	−0.905810
$962$	63.2330	2.03871
$963$	0	0
$964$	5.90867	0.190305
$965$	0	0
$966$	0	0
$967$	−22.8453	−0.734654	−0.367327	−	0.930092i	$-0.619727\pi$
$967$	−22.8453	−0.734654	−0.367327	+	0.930092i	$0.619727\pi$
$968$	−17.4106	−0.559597
$969$	0	0
$970$	0	0
$971$	−42.3917	−1.36042	−0.680208	−	0.733019i	$-0.738110\pi$
$971$	−42.3917	−1.36042	−0.680208	+	0.733019i	$0.738110\pi$
$972$	0	0
$973$	−10.4880	−0.336230
$974$	−28.8524	−0.924492
$975$	0	0
$976$	−15.3881	−0.492560
$977$	0.0842590	0.00269568	0.00134784	−	0.999999i	$-0.499571\pi$
$977$	0.0842590	0.00269568	0.00134784	+	0.999999i	$0.499571\pi$
$978$	0	0
$979$	−5.84297	−0.186742
$980$	0	0
$981$	0	0
$982$	−32.1409	−1.02566
$983$	−15.0122	−0.478813	−0.239407	−	0.970919i	$-0.576953\pi$
$983$	−15.0122	−0.478813	−0.239407	+	0.970919i	$0.576953\pi$
$984$	0	0
$985$	0	0
$986$	−2.39904	−0.0764011
$987$	0	0
$988$	−9.44339	−0.300434
$989$	5.92170	0.188299
$990$	0	0
$991$	3.36536	0.106904	0.0534521	−	0.998570i	$-0.482978\pi$
$991$	3.36536	0.106904	0.0534521	+	0.998570i	$0.482978\pi$
$992$	−4.45185	−0.141346
$993$	0	0
$994$	24.4137	0.774357
$995$	0	0
$996$	0	0
$997$	1.94573	0.0616220	0.0308110	−	0.999525i	$-0.490191\pi$
$997$	1.94573	0.0616220	0.0308110	+	0.999525i	$0.490191\pi$
$998$	39.5130	1.25076
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.ca.1.4	✓	9
3.2	odd	2		6525.2.a.cc.1.6	yes	9
5.4	even	2		6525.2.a.cd.1.6	yes	9
15.14	odd	2		6525.2.a.cb.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.4	✓	9	1.1	even	1	trivial
6525.2.a.cb.1.4	yes	9	15.14	odd	2
6525.2.a.cc.1.6	yes	9	3.2	odd	2
6525.2.a.cd.1.6	yes	9	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23642 q^{2} -0.471260 q^{4} +3.27673 q^{7} +3.05552 q^{8} +2.30259 q^{11} -5.57932 q^{13} -4.05142 q^{14} -2.83539 q^{16} +1.94031 q^{17} -3.59158 q^{19} -2.84698 q^{22} +1.66263 q^{23} +6.89840 q^{26} -1.54419 q^{28} +1.00000 q^{29} +1.70877 q^{31} -2.60530 q^{32} -2.39904 q^{34} +9.16633 q^{37} +4.44071 q^{38} -8.54038 q^{41} +3.56165 q^{43} -1.08512 q^{44} -2.05571 q^{46} -11.5640 q^{47} +3.73698 q^{49} +2.62931 q^{52} -9.66312 q^{53} +10.0121 q^{56} -1.23642 q^{58} -9.83385 q^{59} +5.42714 q^{61} -2.11276 q^{62} +8.89204 q^{64} -5.20114 q^{67} -0.914391 q^{68} -6.02596 q^{71} -15.5143 q^{73} -11.3334 q^{74} +1.69257 q^{76} +7.54498 q^{77} +12.2465 q^{79} +10.5595 q^{82} -10.7179 q^{83} -4.40370 q^{86} +7.03562 q^{88} -2.53756 q^{89} -18.2820 q^{91} -0.783531 q^{92} +14.2980 q^{94} +5.89998 q^{97} -4.62048 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38}+ \cdots - 51 q^{98}+O(q^{100}) \) 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8 + 2 * q^11 - q^13 - 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 - 3 * q^22 - 16 * q^23 + 6 * q^26 + 4 * q^28 + 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 - 10 * q^41 - 3 * q^43 - 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 + 9 * q^52 - 22 * q^53 + 22 * q^56 - 2 * q^58 + 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 - 5 * q^67 - 39 * q^68 + 10 * q^73 - 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 + 8 * q^82 - 46 * q^83 + 28 * q^86 - 2 * q^88 + 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 - 7 * q^97 - 51 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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